(Chip Somodevilla/GETTY IMAGES)

Sen. Rand Paul, a Republican freshman from Kentucky who has previously sparred with leadership on other measures, said in a brief interview that while he will likely oppose the debt-limit plan outlined by Senate Minority Leader Mitch McConnell (R-Ky.) Sunday morning, he will not seek to push a Senate vote on the measure past the deadline by which Congress must act or else the country will default.

“I’ve sort of said all along that I can’t vote for any plan that doesn’t balance at some finite time, and this, to my understanding, never balances,” Paul said.

Asked whether he might seek to delay Senate consideration of the measure, he responded: “Well, I think there’ll be quite a few people that will insist that it be a 60-vote margin, but I would probably agree to let the time waive.”

The Hill reported Sunday that Sen. Mike Lee (R-Utah), another conservative freshman, said that he, too, would not seek to hold up the vote past the deadline.

“I don’t see any reason for not agreeing to an advanced time table,” Lee told The Hill. “A 60-vote threshold is one thing but there’s no reason to run the clock out any more.”

A Lee spokesman said Sunday that the Utah Republican would likely oppose the plan.

“If it just includes just a vote on a (balanced budget amendment) and does not require passage of a BBA, he will not support it,” Lee’s communications director, Brian Phillips, said. “Those of us who signed the cut, cap, balance pledge are committed to that.”

Several other Senate conservatives, including Sens. Jim DeMint (R-S.C.) and Marco Rubio (R-Fla.), did not appear eager to push the debt-ceiling vote to the limit, The Hill reported.

The statements by the Senate conservatives are the latest sign that Congress may be able to pass a debt deal and avoid defaulting on the country’s debt obligations.

Even so, the timing of a Senate vote on a potential deal remains in flux.

Senate Majority Leader Harry Reid (D-Nev.) suggested a vote was still possible Sunday. Asked if he would give long notice before calling a vote, Reid said he would do so but warned: “Don’t go to a ballgame.”

Several senators of both parties, however, projected Sunday that a debt vote would not take place by the end of the day.

“There’ll be no votes of significance whatsoever today, in my opinion,” Sen. Bob Corker (R-Tenn.) said. “They’re very close, only a few details to be worked out, and so again, it is just best to wait until that happens. ... Does it start in the House? Does it start in the Senate? You know, all those kind of things. I think that part hasn’t been fully thought through yet. I think they’re still trying to work on what the solution is first.”

Sen. Orrin Hatch (R-Utah) predicted that “it’ll have to have a cloture vote probably Tuesday morning,” although he cautioned that he didn’t know what leaders were planning.

The Senate was in recess following its vote Sunday afternoon to table Reid’s debt-limit proposal. Senate Republicans had been expected to meet at 1:45 p.m., but the meeting was called off as the details of a debt deal were still being worked out, several members said. Senate Democrats, too, had been slated to meet at 2 p.m. but that meeting was postponed.

Still, closed-door meetings were underway across the Capitol.

Senate Minority Whip Jon Kyl (R-Ariz.) and Sen. Rob Portman (R-Ohio), a former White House budget director, were seen entering Kyl’s office after votes. Meanwhile, Reid, Senate Majority Whip Dick Durbin (D-Ill.) and Sen. Charles Schumer (R-N.Y.) – the Senate’s top three Democrats – huddled in House Minority Leader Nancy Pelosi’s (D-Calif.) office Sunday afternoon; Reid told reporters again that he was “cautiously optimistic” about a deal.

The House was only briefly in session for a “pro forma” session during which no legislative business was conducted. House Republicans were expected to hold a conference call for members later Sunday afternoon on which leaders would brief them on the details of the emerging debt deal.

As the clock ticked closer to Aug. 2, senators almost unanimously expressed optimism that a deal could be passed through Congress by the Tuesday deadline and that an emergency short-term measure would not be necessary.

“Oh yeah,” Corker told reporters when asked whether a deal would be reached by Tuesday. “Yeah, yeah, yeah, yeah. Absolutely. ... I think leadership still expresses optimism that by Tuesday a deal will happen, and they’ll be able to say enough in the way of positive comments that the markets hopefully will see that something’s going to happen.”

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